F. Riesz Theorem on harmonic and subharmonic functions In the book "Uniform Algebras and Jensen Measures" by T.W. Gamelin, p.39, says: 
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By the F.Riesz Theorem, any subharmonic function $u$ in a neighborhood of a compact set $K$ in $C$ can be expressed in the form
$$u(z)= v(z) + \int log|z- \zeta|d\tau(\zeta),\quad z \in K$$
where $\tau$ is a positive measure supported on a compact neighborhood of $K$, and $v$ is harmonic in a neighborhood of $K$.
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Where can I find a reference for this result, especially for a proof? Can anyone provide a proof? Thank you.
 A: A proof for the fundamental theorem on subharmonic functions:
https://projecteuclid.org/download/pdf_1/euclid.tmj/1178245415
A: This question was posed a long time ago so I will answer it from another point of view.
If you're willing to accept some things about distributions you can see this by looking at the potential as a convolution: if $\tau$ is seen as a distribution with compact support and then we have
\begin{equation}
    p_\tau(z) = \int \log |z - w|\ d\tau(w) = \log|z|*\tau
\end{equation}
Now using the fact that $\Delta\log|z| = 2\pi\delta$, we have that $\Delta p_\tau = \Delta(\log|z|*\tau) = 2\pi(\delta*\tau) = 2\pi\tau$.
We need to accept some facts: 


*

*For a subharmonic function $f$ we have $\Delta f$ is a positive distribution.

*Positive distributions come from positive Radon measures (also see this ).

*$\Delta$ is a hypoelliptic operator meaning that if $\Delta h = 0$ for any distribution $h$, then $h$ is a harmonic (classical) function.


If we accept this, then the result follows since $\tau=\frac{1}{2\pi}\Delta f$ is a radon measure and $h = f - p_\tau$ is a harmonic function such that
\begin{equation}
    f = h + p_\tau.
\end{equation}
PS: We also have to check that $\operatorname{supp} \tau$ is compact, so you actually need to change $f$ a little, but it is not too problematic.
PSS: I think you wanted a more classical answer but since 2 years had passed, i figured why not.
